Episode 2 Supports
Exploring: Keoni and Sasha use points that share an x-value to explain why increasing the p-value in the equation y = x2/(4p) results in the parabola getting wider.
Students’ Conceptual Challenges
When asked to justify why the parabolas get wider as the p-value increases, Sasha restates her claim. She and Keoni struggle to identify evidence that can support their claim [2:11-2:19].
For use in a classroom, pause the video and ask these questions:
1. [Pause video at 0:53]. How can we confirm that the three labeled points are on each parabola?
2. [Pause video at 3:02]. What do you think that Keoni means when he says “that it is getting wider by staying…”?
Invite students to engage in stating and justifying mathematical claims with another student:
1. Find the coordinates of points on each of the three parabolas when the x-value is 1.
2. Considering the ordered pairs that you found, what do you notice about the y-values when the p-value increases? How does that impact the shape of the parabola?
Mathematics in this Lesson
CC Math Standards
CC Math Practices
Sasha and Keoni use algebraic and geometric thinking to form three arguments that justify why a parabola gets wider on the coordinate grid as the p-value in y = x2/(4p) increases.
This lesson can help students:
Common Core Math Standards
• CCSS.M.HSF.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Sasha and Keoni first notice key features of the graphs of three quadratic functions (all with a vertex at the origin but with different p-values). For example, they notice that when the x-value is the same for all three functions (e.g., when x = 2), then the y-values of the points decrease as the distance between the vertex and focus (the p-value) increases. They observe visually on the graph how decreasing the y-value but keeping the x-value constant forces the parabola to become wider. They then link this observation to the algebraic expression for the parabolas. By fixing the x-value as 2, the equation y = x2/(4p) becomes y = 4/(4p), which is y = 1/p. Sasha and Keoni can see algebraically that as p increases, y decreases. Comparing the algebraic and graphical representations of these quadratic functions provides a way to help understand why increasing the p-value in y = x2/(4p) means the parabolas will appear to be getting wider on the same coordinate grid.
Common Core Math Practices
CCSS.Math.Practice.MP7. Look for and make use of structure.
In this lesson, Sasha and Keoni construct three different viable arguments to explain why increasing the p-value in y = x2/(4p) means the parabolas will appear to be getting wider on the same coordinate grid. Each argument relies on a consideration of different features of the graphs: (a) fixing the x-value in Episodes 2-3; (b) fixing the y-value in Episode 4; and (c) considering special points in Episodes 5-6. Noticing a relationship is much easier than creating an argument to explain why the relationship holds! To accomplish the latter, Sasha and Keoni express their ideas to each other [Episode 2, 2:33 – 3:42], even when it’s difficult to do so. They also justify their conclusions [Episode 2, 5:02 – 5:33] and respond to each others’ explanations [Episode 6, 1:26 – 1:56].